A328856 Number of factorizations of n into distinct numbers with an odd number of distinct prime factors.
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 2
Offset: 1
Keywords
Examples
a(32) = 3 because 32 = 4 * 8 = 2 * 16.
Links
Programs
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PARI
seq(n)={my(v=vector(n, k, k==1)); for(k=2, n, if(omega(k)%2, my(m=logint(n, k), p=(1 + x + O(x*x^m)), w=vector(n)); for(i=0, m, w[k^i]=polcoef(p, i)); v=dirmul(v, w))); v} \\ Andrew Howroyd, Oct 29 2019, In older versions of PARI, use polcoeff instead of polcoef. - Antti Karttunen, Oct 29 2019 -
PARI
A328856(n, k=n) = (((n<=k)&&((1==n)||(omega(n)%2))) + sumdiv(n, d, if(d > 1 && d <= k && d < n && (omega(d)%2), A328856(n/d, d-1)))); \\ Antti Karttunen, Oct 29 2019
Formula
Dirichlet g.f.: Product_{k>=1} (1 + A030230(k)^(-s)).
a(n) <= A045778(n). - Antti Karttunen, Oct 29 2019
Extensions
More terms from Antti Karttunen, Oct 29 2019